The title of this post is a famous question (posed, for example, by Joe Polchinski) which is modeled after an even more famous question by Ken Wilson, “What is Quantum Field Theory?”. I certainly can’t answer the first question, but Wilson’s question now does have a widely agreed upon answer (which is sadly not well presented in a popular literature that continues to repeat old myths about regularization) which I will mention a bit later

What I would mainly like to do, however, is to answer the much easier question, “What is string perturbation theory?” But before getting to that, let’s talk a bit about what perturbation theory is. Unfortunately, most of the equations in classical and quantum physics are not exactly solvable. Often, one can try to solve them on computers, but, particularly in the case of quantum field theory, the calculations can be long and difficult when they are even possible. Faced with this, most of physics (and other fields) is done by a variety of approximation techniques. The most prevalent of these is perturbation theory.
The idea behind perturbation theory is to start with a set of equations you can solve. You can then “perturb” these equations by adding terms with small coefficients. One expects that the solution to these perturbed equations should be close to the solutions of the unperturbed equations. (This is not always the case, and determining when one’s approximations are valid is an essential part of understanding the physics.) The solutions one obtains via perturbation theory are what are called formal power series — the “formal” just means that they don’t necessarily have to converge. By looking at the first few terms, even if the full series does not converge, one can usually obtain a good approximation to the exact solution. It’s not ideal — so called “non-perturbative” effects can be extremely important, and there’s often not a small parameter to let you even start your perturbation theory — but sometimes it’s the best you can do. For example, the determination of the “g-2″ of the electron is mostly done using perturbation theory and is accurate to 12 or so digits.

Perturbation theory is the connection between quantum field theory and the idea of particles. As you might guess from it’s name, quantum field theory (QFT) is a theory of fields. At every point in spacetime, these fields can take their values, and we have to integrate over every possible way that can happen. Exact solutions are few and far between. So, we do perturbation theory. The easiest exact solutions are solutions to what are called free field theories. Free field theories have the property that the solutions can be expressed in terms of fundamental solutions which we can think of as particles. The difference between this and quantum mechanics is that quantum mechanics deals with a finite fixed number of particles, and free quantum field theory has as many as you want. This is why quantum field theory is sometimes considered as multibody quantum mechanics.

We can do perturbation theory around these free solutions. Because our solutions are close to the solution of the free theory, they can be thought of in terms of particles. The difference is that the particles can now interact with each other — an interacting theory is another name for a QFT which isn’t free. Feynman discovered an amazing and intuitive way of organizing the calculations involved in perturbation theory. Each term is a sum of graphs where the lines represent different types of particle in the free theory and the vertices correspond to the terms which describe the interaction. These are the famous Feynman diagrams:

These diagrams are remarkably useful as it’s very easy to picture them as particles interacting with each other. It’s always worth remembering, however, that they are tools for organizing an approximation. It’s unfortunate, however, that if you actually try to do the calculation the Feynman diagram is pictorially representing, you almost always get infinity. For a long time these infinities bedeviled theorists and were dealt with by a combination of black magic and handwaving. These days, however, we realize that these infinities are reflecting the fact that the theories we are working with are what are called “effective field theories”. This means that they are fundamentally incomplete, only capable of describing part of a world rather than all of it. The infinite answers that appear reflect the incompleteness of the theory, and we can use that insight to find the part of the answer that corresponds to the physics the theory does describe. The previous black magic was actually secretly doing this procedure, and that’s why it gave the correct answer.

Returning to perturbation theory, Feynman diagrams were far from Feynman’s only contribution to quantum field theory. He also showed how to take the fields out of free field theories. Instead of considering fields, one considers all possible ways a particle can move in spacetime. Mathematically, this corresponds to a theory that lives on a one dimensional line as opposed to on spacetime. This line is called the ‘worldline’ of the particle. The important field that lives on this one dimensional line is a map that embeds the line into spacetime (where we’d ordinarily be doing our physics). This values of this map give usual worldline of a particle as show here:

Because we’re doing quantum field theory on this one dimensional line, we integrate over every possible embedding. This gives the same answers as a free field theory in spacetime. It’s a weird inversion; instead of physics living in spacetime, we have that spacetime lives on this one dimensional line. But it works. We can extend this to perturbation theory by adding in the interactions by hand. We have graphs that embed into spacetime, and the same procedure reproduces the results of perturbation theory. You don’t get all of quantum field theory, but it’s still pretty cool.

String perturbation theory is a generalization of this final construction. All we do is replace our one dimensional line with a two dimensional surface called the worldsheet. Two dimensions is a lot more than one, so we have some freedom about the type of theory that can live there. This leads to the various types of string theories, ie, the bosonic string and the various superstring theories. Some amazing things begin to happen once you fix your theory, however. For one, you don’t have to put in interactions by hand anymore. If you look at every possible two dimensional shape to map into spacetime, you automatically get interactions. A sphere with three holes, for example, corresponds to two strings coming in and one string coming out in a diagram often called a “pair of pants”:

Another consequence is that not every spacetime you try to embed your string into ends up giving a sensible theory. Some of the classical theories (which always make sense) don’t have corresponding quantum theories (this is called an anomaly). In particular, the number of dimensions of spacetime is fixed in the simplest examples (there are other theories called non-critical strings which can have varying dimensions, but they tend to exhibit odd behavior.). It also turns out that the spacetime you map into is forced to obey the Einstein field equations. In other words, string perturbation theory only makes sense when you are doing perturbations around a spacetime that satisfies the equations of gravity.

This isn’t a surprise. When you look at how the vibrations of the string manifest themselves as particles in spacetime, one of them looks exactly like a graviton. On reasonably general grounds, any sensible theory that contains a graviton pretty much has to be Einstein’s theory. So, it seems that a minor miracle has occurred. Solely by generalizing Feynman’s description of perturbation theory from one dimensional objects to two dimensional objects, you automatically get gravity. It’s unavoidable. Still, you might object that this putative graviton could only look like a graviton — how do we know it actually is one? A graviton is supposed to be a small perturbation in the metric of spacetime. So, we can write down a theory on the two dimensional worldsheet mapping into a space with one of these perturbed metrics. By doing the usual tricks of perturbation theory, we can convert this into a state of the theory with the unperturbed metric. This state turns out to be precisely a bunch of our putative gravitons. Thus, we see that combining a ton of these gravitons precisely corresponds to changing the metric, and we can drop the “putative”.

That’s string perturbation theory. Just like in quantum field theory, the power series you get doesn’t converge. However, while it hasn’t been completely rigorously proven, there are very good arguments that, unlike in quantum field theory, every individual term you get is finite. This means that string theory doesn’t have to be an effective theory — it’s a possible theory of everything.

Well, it would be, except that it’s a perturbation theory. It’s only an approximation, and we don’t know to what. So, that’s the best I can do to answer the title of this post: string theory is the theory that has string perturbation theory as its perturbative approximation. Not very helpful, I realize, but in the last decade and a half of string theory, we’ve been able to learn a lot about the properties of this conjectural theory. We’ve even managed to give complete definitions in spacetimes that obey certain boundary conditions. But, it is still this puzzle that remains the fundamental question of string theory: we don’t know what it is.

Comments

Aaron, thanks for this post. In an effort to rectify common misconceptions let me add something about convergence of the series expansions appearing in perturbation theories. My point is twofold- first such series do not have to converge, secondly if your want your results to be interesting and realistic they better not converge.

On the first point, if the perturbation series converges for all quantities then the sums are well defined and therefore the theory is solvable. For theories that are not integrable (= solvable) this does not happen, but it does not mean the series is useless. It only means that there is a lower bound on the error in the perturbative estimate of the real answer. However, that error can be very low for all practical purposes, depending on values of parameters, but it is always there and therefore the series cannot converge.

On the second point- once again if the series converges it means the theory is very special- it is integrable which means it has many conservation laws etc., and in particular it means it bars a lot of the interesting physics (chaos, ergodicity,…) from happening. If you are interested in theory of everything, or of anything interesting really, you are not looking for an integrable model, and you will have to deal with divergent series.

(not that integrable models are not useful as toy models, but that is a different question).

It depends what you calculate, in the simplest case you just calculate a number, say the anomalous magnetic moment of the electron. The theory has an answer to that (which may or may not agree with experiment) but PT only gives an approximation to that answer with an error that is bounded from below. For the above mentioned quantity in QED this lower bound is ridiculously small so you are not too worried about it. Consequently the fact that PT does not converge is not nearly the disaster it naively appears to be.

On reasonably general grounds, any sensible theory that contains a graviton pretty much has to be Einstein’s theory.

How so?

I mean, why is it that the presence of a graviton implies it is an Einsteinian field / relativity theory?

Also: When we say that string theory has a graviton, isn’t what we really mean that it has a spin-2 massless particle? It seems reasonable to say that if such a particle exists we can say it’s the graivton, but going all the way from “the theory predicts a spin-2 massless particle” to “the theory must follow relativity”… it seems like there’s one leap too far in that somewhere. Am I missing something?

So, it seems that a minor miracle has occurred. Solely by generalizing Feynman’s description of perturbation theory from one dimensional objects to two dimensional objects, you automatically get gravity.

I’m confused about this too. Surely the emergence of gravity you describe here occurs because of the principles of string theory, not because you generalized perturbation theory to the two dimensional worldsheet? Or are you saying that string theory is somehow an inherent consequence of using worldsheets?

Hi Aaron, glad to see you’ve been elevated to Guest Blogger! Needless to say, you’re more than deserving of this title…

Not to come across as a suck-up, but you’re one of my favorite commenters during the height of the String Wars. And not to sound overly biased, but to me, you deserve most (if not all) of the credit for coining the phrase, “String Wars”.

Be mindful, I’m far from being describes as a physicist, let alone a string theorist, so please don’t be too hard on my naive question… Is it somewhat accurate to say that perturbation theory is to string cosmology as non-perturbation theory is to brane cosmology?

“It’s unfortunate, however, that if you actually try to do the calculation the Feynman diagram is pictorially representing, you almost always get infinity. For a long time these infinities bedeviled theorists and were dealt with by a combination of black magic and handwaving.”

Did you know that in qubits, Feynman diagrams are finite to all orders in perturbation theory? See arXiv:0705.2121 . If you want to explain the (point) mass interaction between left and right handed particles, this gives you a kind and gentle way of doing it.

I’m not sure where this falls on the continuum between folk theorem and physics theorem, but the general idea is that to quantize an interacting massless spin-2 field, you have to impose a gauge invariance. That gauge invariance then leads you to writing down curvature invariants, the lowest order of which (other than the cosmological constant) is the Einstein-Hilbert lagrangian.

it seems like there’s one leap too far in that somewhere. Am I missing something?

I tried to explain this after the sentence starting with “Still, you might object”.

Surely the emergence of gravity you describe here occurs because of the principles of string theory, not because you generalized perturbation theory to the two dimensional worldsheet? Or are you saying that string theory is somehow an inherent consequence of using worldsheets?

I don’t know what the principles of string theory are. The point of this post is that I do know what string perturbation is, and it’s a generalization of Feynman’s description of perturbation theory for QFT to worldsheets. Just from that, gravity automatically appears. Whatever string theory is, it better have string perturbation theory as its perturbation expansion.

Is it somewhat accurate to say that perturbation theory is to string cosmology as non-perturbation theory is to brane cosmology?

I’d say this instead. D-branes allow us to get a handle on some nonperturbative effects in string theory and thus go beyond string perturbation theory.

It’s The Emperor’s New Clothes. The Emperor is everything, its a singularity, if you can’t see it’s nakedness then you will never escape from its pull, to see its nakedness you will have to prove that String Theory is wrong.

A little late in commenting, but I second the thanks on this post. As my studies stopped after QM and special relativity, string theory is “twice removed”, so one has to read simplified accounts several times to “get” what you can’t practice on.

For one, you don’t have to put in interactions by hand anymore.

I can see how that works in the case of mediating particles and to the exclusion of other diagrams which I then gather are unphysical. Nice!

Btw, I’m reading Brain Greene’s popular text The Elegant Universe, and flipping ahead he uses the motivation that these “pants diagrams” smooths out spacetime singularities from particle diagrams more definitively placed events.

if the series converges it means the theory is very special- it is integrable which means it has many conservation laws etc., and in particular it means it bars a lot of the interesting physics (chaos, ergodicity,…) from happening.

But I don’t see that this necessarily follows. Chaos could be classical, couldn’t it?

That was just an example, but absolutely, you can use perturbation theory in classical mechanics, or in any context you are solving a differential equation. The series you get almost always do not converge, and that is a good thing.

Regarding Moshe’s comment:
“On the second point- once again if the series converges it means the theory is very special- it is integrable which means it has many conservation laws etc., and in particular it means it bars a lot of the interesting physics (chaos, ergodicity,…) from happening. If you are interested in theory of everything, or of anything interesting really, you are not looking for an integrable model, and you will have to deal with divergent series.”
I beg to differ a little on the interpretation. In such cases as approximations to classical physics, which is an integrable theory, ergodicity and chaos arise naturally, it is a “bottom-up” approximation. You can go even further. For example, in loop quantum cosmology one assumes a high degree of symmetries and through this approximation attains some pretty interesting results as well. String theory goes the opposite way, but to me it seems nothing guarantees there is anything waiting at the other side!

Classical physics is not “integrable”: for most systems one cannot write a general closed form expression for the trajectories (e.g. as a convergent power series), which is what it means to “integrate” a classical system.

(my comments have nothing to do with string theory or LQC, no need to go there).

wow! that was by far my favourite post on the subject of string theory. thanks for being so clear and lucid. this post really puts the theory into context for me. a thoroughly fascinating read! look forward to hearing more from you.

So, we can write down a theory on the two dimensional worldsheet mapping into a space with one of these perturbed metrics. By doing the usual tricks of perturbation theory, we can convert this into a state of the theory with the unperturbed metric. This state turns out to be precisely a bunch of our putative gravitons. Thus, we see that combining a ton of these gravitons precisely corresponds to changing the metric

So this would mean that the state of the graviton field determines the metric and therefore what the geodesic that is the classical path is. But how is it determined that the graviton couples to mass, in such a way so as to produce Einstein’s feild equations?

Also its not quite clear to me what is meant by the string perturbation theory in a curved metric being the same as a with a flat metric and some gravitons. Does this mean that a certain state of the graviton field gives a certain metric, which seems at odds with the indeterminant nature of the stress-energy tensor, or is the graviton description somehow equivalent to an indeterminate spacetime due to the pertubative nature of how this relationship was determined? Or am I just completely missunderstanding this?

Also what does this curvature mean for the extra curled up dimmensions?

Moshe, thanks for your answer! I take it non-integrability isn’t required for a fundamental theory then. (But it is more interesting if it exists.)

Because I have a hard time wrapping my head around the difference between dissipation which gives classical non-integrability, and a fundamental theory similar loss of… what?

Not probability, because that would mean non-unitarity, wouldn’t it? Information, perhaps – “one cannot write a general closed form expression for the trajectories” would apply in any perturbation theory AFAIU the post.

I would be a little more careful and say that the state of the graviton determines a slightly perturbed metric. You can work out the couplings of the graviton state to other particles by doing scattering calculations and see that it does couple in the correct manner.

I’m not sure I understand the questions in your second paragraph, but if you think of everything as small perturbations (as GR perturbation theory) perhaps it might help. The curled up extra dimensions can have their curvature perturbed, too, and those appear as particles (called Kaluza-Klein modes) in the large dimensions we observe. Not observing types of these particles is a strong constraint on the nature of the curled up dimensions.

Aaron, I’m confused, how do you calculate scattering without knowing the coupling?

I’ll try to rephrase my other question. The way you explained the equivalence of gravitons and a perturbed metric, made me think that a specific state of gravitons would lead to a specific curvature. However I’m sure that is not what you meant since there can’t possibly be a precise curvature when the stress energy tensor is somewhat indeterminate. I was hoping you could clear this up for me.

I’m not sure what you mean by saying that the stress energy tensor is indeterminate. The scattering is calculated using the rules of string perturbation theory which, as I said in the text, does not require the specification of couplings — they’re already there in the pants diagram. Everything else is as in GR perturbation theory.

Books

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